Signal analysis on the sphere

在球体表面测量数据的领域多种多样，如计算机图形学、计算机视觉、地球物理学、行星科学、分子生物学、声学和天体物理学，仅举几例。一旦对方向进行了观测，得到的数据自然就存在于球体上。然而，迄今为止，大多数信息学和信号处理技术都局限于欧几里得空间。这些信息学技术在工程和物理的许多领域被证明是非常有用的；然而，它们目前还不能应用于球面上定义的大量数据集。为了实现信息学技术在球面数据集上的优势，我们将把欧几里得信息学技术扩展到球面，重点关注三个基本理论和实践重要性的领域：即采样理论、小波变换和解决球面上逆问题的技术。奈奎斯特-香农采样理论是信息论的一个开创性成果，描述了如何从有限数量的样本中捕获带限信号的所有信息内容。从信息论的角度来看，捕获信号信息内容所需的样本数量是采样定理的基本性质。球面上的采样理论不如欧几里得空间上的成熟。最近，McEwen在球上发展了一个新的抽样定理，与之前由Driscoll和Healy在1994年提出的标准抽样定理相比，它将球奈奎斯特率降低了两倍。我们将把这个结果扩展到由旋转群SO(3)定义的三维旋转空间，通常由欧拉角参数化。这将使旋转群上定义的信号的奈奎斯特采样减少两倍。此外，我们将开发快速和精确的算法来计算在旋转群上定义的信号的傅里叶变换，即所谓的Wigner变换。小波是一种强大的信号分析工具，因为它能够同时在尺度和位置上定位信号内容。McEwen最近在球上构造了精确小波变换来对球上定义的标量函数进行定向分析。目前还不存在能够对球体上的自旋信号（如偏振光）进行定向分析的小波变换。我们将构建这样一个小波框架，并在快速维格纳变换的基础上开发快速而精确的算法，将小波变换应用于大的球形数据集。小波变换提供的采样定理和稀疏分解是压缩感知的革命性新范例的基石。在压缩感知中，利用自然信号的稀疏性（在有效表示中），通过解决一个逆问题，从比典型的更少的测量中恢复信号。在这一理论的鼓励下，稀疏正则化技术解决逆问题最近得到了广泛的应用，并显示出相当大的前景。我们将开发一个通用的，灵活的和连贯的框架，通过促进稀疏性，利用我们的新颖的采样理论和小波变换来解决球体上的逆问题。

Data are measured on the surface of a sphere in fields as diverse as computer graphics, computer vision, geophysics, planetary science, molecular biology, acoustics, and astrophysics, to name only a few. As soon as observations are made over directions, the resulting data naturally live on the sphere. However, the majority of informatics and signal processing techniques developed to date are restricted to Euclidean space. These informatics techniques have proved exceptionally useful in many areas of engineering and physics; however, they cannot at present be applied to the large variety of data-sets defined on the sphere. To realise the benefits of informatics techniques on spherical data-sets, we will extend Euclidean informatics techniques to the sphere, focusing on three areas of fundamental theoretical and practical importance: namely, sampling theory, wavelet transforms, and techniques to solve inverse problems on the sphere.The Nyquist-Shannon sampling theory is a seminal result in information theory, describing how to capture all of the information content of a band-limited signal from a finite number of samples. From an information theoretic perspective, the number of samples required to capture the information content of a signal is the fundamental property of a sampling theorem. Sampling theory on the sphere is less mature than in Euclidean space. Very recently McEwen developed a new sampling theorem on the sphere that reduces the spherical Nyquist rate by a factor of two compared to the previous canonical sampling theorem developed by Driscoll & Healy in 1994. We will extend this result to the space of three-dimensional rotations defined by the rotation group SO(3), often parameterised by the Euler angles. This will reduce Nyquist sampling of signals defined on the rotation group by a factor of two. Furthermore, we will develop fast and exact algorithms to compute the Fourier transform of signals defined on the rotation group, the so-called Wigner transform.Wavelets are a powerful signal analysis tool due to their ability to localise signal content in scale and position simultaneously. McEwen recently constructed exact wavelet transforms on the sphere to perform a directional analysis of scalar functions defined on the sphere. At present a wavelet transform capable of performing a directional analysis of spin signals on the sphere, such as polarised light, does not exist. We will construct such a wavelet framework and will develop fast and exact algorithms, based on our fast Wigner transforms, to apply this wavelet transform to big spherical data-sets.A sampling theorem and sparse decompositions like those afforded by a wavelet transform are the building blocks of the revolutionary new paradigm of compressive sensing. In compressive sensing, the sparsity of natural signals (in an efficient representation) is exploited to recover a signal from fewer measurements than typical by solving an inverse problem. Encouraged by this theory, sparse regularisation techniques to solve inverse problems have recently found widespread application and shown considerable promise. We will develop a generic, flexible and coherent framework for solving inverse problems on the sphere by promoting sparsity, exploiting our novel sampling theory and wavelet transforms described above.